Suppose that $\displaystyle f$ is entire and that $\displaystyle |f(z)-1|<1$ for all $\displaystyle z$ with $\displaystyle |z|=1$. Prove that $\displaystyle f(z)$ has no zeroes in the disk $\displaystyle \Delta= \{z:|z|<1\} $.

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- Nov 15th 2011, 11:16 AMtarheelbornMaximum Modulus Theorem
Suppose that $\displaystyle f$ is entire and that $\displaystyle |f(z)-1|<1$ for all $\displaystyle z$ with $\displaystyle |z|=1$. Prove that $\displaystyle f(z)$ has no zeroes in the disk $\displaystyle \Delta= \{z:|z|<1\} $.

- Nov 15th 2011, 11:45 AMFernandoRevillaRe: Maximum Modulus Theorem
- Nov 15th 2011, 12:37 PMtarheelbornRe: Maximum Modulus Theorem
We haven't had Rouche's Theorem. I have the Maximum Modulus Theorem and I am not really sure how to apply that here.

- Nov 15th 2011, 12:50 PMxxp9Re: Maximum Modulus Theorem
The maximum modlus theorem states if g is holomorphic, |g| reaches its maximum only on the boundary.

Now g=f-1 is holomorphic, and |g|<1 on |z|=1, so |g|<1 in the disk |z|<1.

So |f|=|1+g|>=|1-|g||>|1-1|=0 in the disk.